Small-slope elastic membrane energy
= Small-slope elastic membrane energy
For a nearly flat membrane represented by height $h(x)$, the quadratic bending-and-tension energy per invariant transverse length is
$$
E=\frac12\int\left(k_ch_{xx}^2+\sigma h_x^2\right)dx.
$$
Its free equilibrium equation is $k_ch_{xxxx}-\sigma h_{xx}=0$.